Angle Relationships

📐 Angle Relationships

Part 1 of 5 — Concept Introduction

Angles are everywhere — in the corner of a book, the hands of a clock, and the spot where two roads cross. In this lesson you'll learn how angles relate to each other so you can find a missing angle without even using a protractor!

Quick Review: What Is an Angle?

An angle is formed by two rays that share a common endpoint called the vertex. We measure angles in degrees ($°$), and one full turn all the way around is $360°$.

We sort angles by how big they are:

Type	Measure	Picture in your head
Acute	less than $90°$	a sharp, narrow corner
Right	exactly $90°$	a perfect square corner
Obtuse	between $90°$ and $180°$	a wide, open corner
Straight	exactly $180°$	a flat straight line

Keep these in mind — the relationships below are built right on top of them. ✅

The Two Big "Sum" Relationships

Two of the most useful angle relationships are about angles that add up to a special number.

Complementary angles are two angles that add up to $90°$.

• $30°$ and $60°$ are complementary because $30 + 60 = 90$.
• $45°$ and $45°$ are complementary because $45 + 45 = 90$.

Supplementary angles are two angles that add up to $180°$.

• $120°$ and $60°$ are supplementary because $120 + 60 = 180$.
• $130°$ and $50°$ are supplementary because $130 + 50 = 180$.

Important: The two angles do not have to be touching! As long as their measures add to $90°$ they are complementary, and as long as they add to $180°$ they are supplementary.

💡 Memory trick: Complementary comes before Supplementary in the alphabet, just like $90$ comes before $180$ on the number line.

Adjacent and Vertical Angles

Adjacent angles sit side-by-side. They share a common vertex and a common side, but they do not overlap. (Think of slicing a pizza wedge into two smaller wedges.)

Vertical angles appear whenever two straight lines cross. The two angles that sit opposite each other (across the X) are vertical angles.

🔑 Key property: Vertical angles are ALWAYS equal!

When two lines intersect they make $4$ angles. Opposite angles are equal, and angles next to each other form a straight line, so they are supplementary.

Worked example: Two lines cross and one angle measures $50°$. The angle opposite it is also $50°$ (vertical angles are equal). The two angles next to it form straight lines, so each is $180 - 50 = 130°$. ✅

Concept Check 🎯

You just met four relationships: complementary, supplementary, adjacent, and vertical. Let's check the key idea.

✏️ Worked Examples: Finding a Missing Angle

Part 2 of 5 — Worked Examples

To find a missing angle, just subtract from the special total.

• To find a complement, subtract from $90°$.
• To find a supplement, subtract from $180°$.

Example 1 — Complement. What is the complement of $35°$?

$90 - 35 = 55$

So the complement of $35°$ is $\mathbf{55°}$. ✅ (Check: $35 + 55 = 90$.)

Example 2 — Supplement. What is the supplement of $110°$?

$180 - 110 = 70$

So the supplement of $110°$ is $\mathbf{70°}$. ✅ (Check: $110 + 70 = 180$.)

Using Algebra with Angle Relationships

Sometimes the angles are written with a variable like $x$. Set up an equation using the correct total, then solve.

Example 3 — Complementary with algebra. Two complementary angles are $3x$ and $2x$. Find each angle.

Step 1 — Write the equation (sum is $90$): $3x + 2x = 90$

Step 2 — Combine like terms: $5x = 90$

Step 3 — Divide both sides by $5$: $x = 18$

Step 4 — Plug back in: $3(18) = 54°$ and $2(18) = 36°$.

The angles are $\mathbf{54°}$ and $\mathbf{36°}$. ✅ (Check: $54 + 36 = 90$.)

Example 4 — Supplementary with algebra. Two supplementary angles are $x$ and $2x$. Then $x + 2x = 180$, so $3x = 180$ and $x = 60$. The angles are $60°$ and $120°$.

Your Turn 🧮

Solve each one. Type just the number (no $°$ symbol).

1. What is the complement of $40°$? ($90 - 40 = ?$)

2. What is the supplement of $75°$? ($180 - 75 = ?$)

3. Two lines cross. One angle is $65°$. What is its vertical angle (the one directly across from it)?

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Work through each question. Pick the relationship first, then do the arithmetic.

Name That Relationship 🔍

Choose the word that correctly completes each statement.

🌍 Real-World Angles

Part 4 of 5 — Application & Word Problems

Angle relationships show up all the time in the real world:

• A skateboard ramp leaning against a wall makes a corner with the floor. The ramp angle and the angle above it on the wall are complementary (they share a right-angle corner that totals $90°$).
• When you fold a piece of paper flat, the angles along the straight crease are supplementary because the fold lies on a straight line ($180°$).
• Where two streets cross in an X-shape, the angles straight across from each other are vertical angles and have equal measures.

Worked example: A kite string makes a $58°$ angle with a flagpole. The pole is straight up, forming a $90°$ corner with the ground. The angle between the string and the ground plus the $58°$ angle must complete that corner, so it is $90 - 58 = 32°$. The string-and-ground angle is $\mathbf{32°}$. ✅

Word Problem Practice 🧮

Type just the number (no $°$ symbol).

1. A door is propped open. The open door and the wall behind it form two supplementary angles. If the door makes a $65°$ angle with the wall on one side, what is the angle on the other side? ($180 - 65 = ?$)

2. A ramp meets the floor at a $25°$ angle. The angle between the ramp and a vertical wall is its complement. What is that angle? ($90 - 25 = ?$)

3. Two roads cross. One of the angles at the crossing is $72°$. What is the measure of the vertical angle across from it?

Apply It 🎯

Read carefully and choose the best answer.

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

You've learned four key angle relationships. Here is everything in one place:

Relationship	What It Means	How to Find a Missing Angle
Complementary	two angles add to $90°$	subtract from $90$
Supplementary	two angles add to $180°$	subtract from $180$
Adjacent	side-by-side, share a vertex & side	depends on the figure
Vertical	opposite angles from crossing lines	they are equal

Remember:

• Complementary → 90° (C comes first, like 90).
• Supplementary → 180°.
• Vertical angles are always equal.

Now put it all together in the final challenge below. 💪

Mixed Challenge 🎯

These questions mix all of today's ideas. Take your time!